This seems a very guarded inscription; it made me curious to know whether there was anything behind it and the more I looked into William Clarence McGregor the more dark thoughts I began to have about him.
His entire eight-eight-page service file has been digitised and for some time it made confusing reading.
The War Graves Commission record says that he was the son of Mrs Jessie McGregor and the late Dugald McGregor and that he served as Murray. According to the documents in his file, he enlisted on 17 September 1914 giving his name as William Clarence McGregor, his birthplace as Bellingen, New South Wales, his profession as motor driver, and his age as 21 and one month. In answer to the question had he ever been apprenticed he answered no. The next document in the file is his discharge paper. There is no information on it, no date of discharge and no information as to why he was discharged.
However, on 2 July 1915, the file contains the attestation form for Albert Murray. There is a note in red ink at the top of the form, 'Real name William Clarence McGregor'. 'Albert Murray' said he was born in Aukland, New Zealand, and that he was a motor mechanic who had been apprenticed for four years to his father in Aukland. In answer to the questions, 'Have you ever been discharged from HM Forces?', 'Have you ever served in HM Forces' and 'Have you ever been rejected as unfit?', his answer to every question was 'no'.
You can see why I was having dark thoughts about McGregor/Murray. Albert Murray received a commission in June 1916, embarked from Australia in January 1917 and served with the 49th Battalion Australian Infantry. However, he didn't get to France until the 17 November that year.
He seems to have been a bold soldier as testified by the manner in which he won his Military Cross on 17 August 1918:

"For conspicuous daring in dealing with a troublesome hostile machine-gun. Crawling over No Man's Land, he entered the enemy's trench & worked up it for about 150 yards, until he located the sentry mounted on the gun. He killed the sentry & captured the gun. After bombing a dug-out & killing an officer & four men, he made good his way back with two prisoners."

Note, citations usually read 'for conspicuous gallantry' not 'daring'. A month later whilst out on patrol he was hit by a machine-gun bullet and killed instantly.
At this point he was still known as Albert Murray. However, a year after his death his mother wrote to the military authorities to say that "as the mother of the above-named soldier, who was killed in action in France on the 16th September 1918, I desire to take the necessary steps to have his correct name recorded". This is the story she had to tell:

"My son enlisted to leave with the first lot of men to go and was very disappointed when he contracted rheumatic fever and instead of sailing with his camp comrades he had to go into hospital for 9 weeks and as a consequence received his discharge.
Later on when he considered that he had removed all trace of the [disease] he endeavoured to re-enlist but was advised that his former illness which had to be disclosed would come against him.
Not to be defeated in this worthy object he enlisted in a name other than his own and sailed as if Lieut Albert Murray in the troopship Ayrshire in 1916 ... "

Mrs McGregor obviously convinced the authorities, which is why his file has 'Correct name William Clarence McGregor' written over all his forms. She also got his correct name carved onto his headstone. However, it's interesting to note that the War Graves Commission told her that they would also include the name under which he served, reasoning:

"If the correct name only appeared in view of the fact that he served under the assumed name there would be danger of his identity being lost sight of."

So, my dark thoughts about McGregor were totally unfounded. His reasons for disguising his identity far from being nefarious were down to the fact that he was keen to join the action and feared that his medical history, if suspected, would prevent him doing so.